Which Is Equivalent To $\log _{\frac{1}{2}} 462$?A. $\frac{\log _{\frac{1}{2}} 462}{\log _{422} \frac{1}{2}}$ B. $ Log ⁡ 462 Log ⁡ 1 2 \frac{\log 462}{\log \frac{1}{2}} L O G 2 1 ​ L O G 462 ​ [/tex] C. $\frac{\log \frac{1}{2}}{\log 462}$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore the concept of logarithmic equations and provide a step-by-step guide on how to solve them. We will also examine a specific problem, which is equivalent to $\log _{\frac{1}{2}} 462$, and provide solutions to three different options.

What are Logarithmic Equations?

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that involves a variable in the exponent of a number. For example, the equation $\log _a x = y$ is equivalent to the equation $a^y = x$.

Properties of Logarithms

There are several properties of logarithms that are essential to solving logarithmic equations. These properties include:

• Product Rule: $\log _a (xy) = \log _a x + \log _a y$
• Quotient Rule: $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$
• Power Rule: $\log _a (x^y) = y \log _a x$
• Change of Base Formula: $\log _a x = \frac{\log _b x}{\log _b a}$

Solving Logarithmic Equations

To solve a logarithmic equation, we need to isolate the variable. This can be done by using the properties of logarithms. Here are the steps to solve a logarithmic equation:

1. Isolate the logarithm: Move all terms except the logarithm to the other side of the equation.
2. Use the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base formula to simplify the equation.
3. Solve for the variable: Once the equation is simplified, solve for the variable.

Which is Equivalent to $\log _{\frac{1}{2}} 462$?

Now, let's examine the specific problem, which is equivalent to $\log _{\frac{1}{2}} 462$. We will provide solutions to three different options.

Option A: $\frac{\log _{\frac{1}{2}} 462}{\log _{422} \frac{1}{2}}$

To solve this option, we need to use the change of base formula. The change of base formula states that $\log _a x = \frac{\log _b x}{\log _b a}$. In this case, we can rewrite the equation as:

$$\frac{\log _{\frac{1}{2}} 462}{\log _{422} \frac{1}{2}} = \frac{\log _{\frac{1}{2}} 462}{\log _{\frac{1}{2}} 422}$$

Using the change of base formula, we can simplify the equation as:

$$\frac{\log _{\frac{1}{2}} 462}{\log _{\frac{1}{2}} 422} = \log _{422} 462$$

This is not the correct solution, as we need to find the equivalent of $\log _{\frac{1}{2}} 462$.

Option B: $\frac{\log 462}{\log \frac{1}{2}}$

To solve this option, we need to use the change of base formula. The change of base formula states that $\log _a x = \frac{\log _b x}{\log _b a}$. In this case, we can rewrite the equation as:

$$\frac{\log 462}{\log \frac{1}{2}} = \log _{\frac{1}{2}} 462$$

This is the correct solution, as we have found the equivalent of $\log _{\frac{1}{2}} 462$.

Option C: $\frac{\log \frac{1}{2}}{\log 462}$

To solve this option, we need to use the change of base formula. The change of base formula states that $\log _a x = \frac{\log _b x}{\log _b a}$. In this case, we can rewrite the equation as:

$$\frac{\log \frac{1}{2}}{\log 462} = \log _{462} \frac{1}{2}$$

This is not the correct solution, as we need to find the equivalent of $\log _{\frac{1}{2}} 462$.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the properties of logarithms. By using the product rule, quotient rule, power rule, and change of base formula, we can simplify logarithmic equations and solve for the variable. In this article, we examined a specific problem, which is equivalent to $\log _{\frac{1}{2}} 462$, and provided solutions to three different options. We found that option B, $\frac{\log 462}{\log \frac{1}{2}}$, is the correct solution.

References

• Logarithmic Equations: A comprehensive guide to logarithmic equations, including properties and examples.
• Change of Base Formula: A formula that allows us to change the base of a logarithm.
• Product Rule: A rule that states $\log _a (xy) = \log _a x + \log _a y$.
• Quotient Rule: A rule that states $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$.
• Power Rule: A rule that states $\log _a (x^y) = y \log _a x$.

Further Reading

• Logarithmic Functions: A comprehensive guide to logarithmic functions, including properties and examples.
• Exponential Functions: A comprehensive guide to exponential functions, including properties and examples.
• Trigonometric Functions: A comprehensive guide to trigonometric functions, including properties and examples.

Glossary

• Logarithm: The inverse operation of exponentiation.
• Exponentiation: The operation of raising a number to a power.
• Change of Base Formula: A formula that allows us to change the base of a logarithm.
• Product Rule: A rule that states $\log _a (xy) = \log _a x + \log _a y$.
• Quotient Rule: A rule that states $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$.
• Power Rule: A rule that states $\log _a (x^y) = y \log _a x$.

FAQs

• Q: What is a logarithmic equation? A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation.
• Q: How do I solve a logarithmic equation? A: To solve a logarithmic equation, you need to isolate the logarithm and then use the properties of logarithms to simplify the equation.
• Q: What is the change of base formula? A: The change of base formula is a formula that allows us to change the base of a logarithm.
• Q: What is the product rule? A: The product rule is a rule that states $\log _a (xy) = \log _a x + \log _a y$.
• Q: What is the quotient rule? A: The quotient rule is a rule that states $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$.
• Q: What is the power rule? A: The power rule is a rule that states $\log _a (x^y) = y \log _a x$.

Appendix

• Logarithmic Equations: A comprehensive guide to logarithmic equations, including properties and examples.
• Change of Base Formula: A formula that allows us to change the base of a logarithm.
• Product Rule: A rule that states $\log _a (xy) = \log _a x + \log _a y$.
• Quotient Rule: A rule that states $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$.
• Power Rule: A rule that states $\log _a (x^y) = y \log _a x$.

Index

• Logarithmic Equations: A comprehensive guide to logarithmic equations, including properties and examples.
• Change of Base Formula: A formula that allows us to change the base of a logarithm.
• Product Rule: A rule that states $\log _a (xy) = \log _a x + \log _a y$.
• Quotient Rule: A rule that states $\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$.
• Power Rule: A rule that states $\log _a (x^y)
  Logarithmic Equations Q&A ==========================

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that involves a variable in the exponent of a number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithm and then use the properties of logarithms to simplify the equation. Here are the steps to solve a logarithmic equation:

1. Isolate the logarithm: Move all terms except the logarithm to the other side of the equation.
2. Use the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base formula to simplify the equation.
3. Solve for the variable: Once the equation is simplified, solve for the variable.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows us to change the base of a logarithm. The change of base formula is:

$$\log _a x = \frac{\log _b x}{\log _b a}$$

Q: What is the product rule?

A: The product rule is a rule that states:

$$\log _a (xy) = \log _a x + \log _a y$$

Q: What is the quotient rule?

A: The quotient rule is a rule that states:

$$\log _a \left(\frac{x}{y}\right) = \log _a x - \log _a y$$

Q: What is the power rule?

A: The power rule is a rule that states:

$$\log _a (x^y) = y \log _a x$$

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to identify the base of the logarithm and the value of the logarithm. Then, you can use the formula to change the base of the logarithm.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. In other words, a logarithmic equation is an equation that involves a variable in the exponent of a number, while an exponential equation is an equation that involves a variable as the exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponent and then use the properties of exponents to simplify the equation. Here are the steps to solve an exponential equation:

1. Isolate the exponent: Move all terms except the exponent to the other side of the equation.
2. Use the properties of exponents: Use the product rule, quotient rule, or power rule to simplify the equation.
3. Solve for the variable: Once the equation is simplified, solve for the variable.

Q: What is the difference between a logarithmic function and an exponential function?

A: A logarithmic function is a function that involves a logarithm, while an exponential function is a function that involves an exponent. In other words, a logarithmic function is a function that involves a variable in the exponent of a number, while an exponential function is a function that involves a variable as the exponent.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to identify the base of the logarithm and the value of the logarithm. Then, you can use a graphing calculator or a graphing software to graph the function.

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to identify the base of the exponent and the value of the exponent. Then, you can use a graphing calculator or a graphing software to graph the function.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in real-life situations, such as:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic equations are used to calculate the stress and strain on a material.
• Computer Science: Logarithmic equations are used to calculate the time complexity of an algorithm.

Q: What are some common applications of exponential equations?

A: Exponential equations have many applications in real-life situations, such as:

• Finance: Exponential equations are used to calculate compound interest and investment returns.
• Science: Exponential equations are used to calculate the growth and decay of a population.
• Engineering: Exponential equations are used to calculate the stress and strain on a material.
• Computer Science: Exponential equations are used to calculate the time complexity of an algorithm.

Q: How do I use logarithmic equations in real-life situations?

A: To use logarithmic equations in real-life situations, you need to identify the problem and then use the properties of logarithms to simplify the equation. Here are some steps to use logarithmic equations in real-life situations:

1. Identify the problem: Identify the problem and the variables involved.
2. Use the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base formula to simplify the equation.
3. Solve for the variable: Once the equation is simplified, solve for the variable.

Q: How do I use exponential equations in real-life situations?

A: To use exponential equations in real-life situations, you need to identify the problem and then use the properties of exponents to simplify the equation. Here are some steps to use exponential equations in real-life situations:

1. Identify the problem: Identify the problem and the variables involved.
2. Use the properties of exponents: Use the product rule, quotient rule, or power rule to simplify the equation.
3. Solve for the variable: Once the equation is simplified, solve for the variable.

Conclusion

In conclusion, logarithmic equations and exponential equations are essential concepts in mathematics and have many applications in real-life situations. By understanding the properties of logarithms and exponents, you can use logarithmic equations and exponential equations to solve problems in finance, science, engineering, and computer science.